Standard Equation of a Circle

A circle is defined as the set (locus) of all points in a plane that lie at a fixed positive distance, called the radius $r$, from a fixed point, called the centre. This purely metric definition -- unlike the focus-directrix definitions used later for the parabola, ellipse and hyperbola -- is what gives the circle's equation its especially simple form.

Derivation. Let the centre be the fixed point $C(h,k)$, and let $P(x,y)$ be any point on the circle. By the definition, the distance $CP$ must equal $r$ for every such point. Using the distance formula between $C(h,k)$ and $P(x,y)$,

Squaring both sides (valid since both sides are non-negative) removes the square root and gives the standard equation of a circle:

Every point satisfying this equation is at distance exactly $r$ from $C(h,k)$, and conversely every point of the circle satisfies it -- so this equation completely characterises the circle, and no other curve.

Centre at the origin. When the centre coincides with the origin, i.e. $h=k=0$, the equation simplifies to

the simplest possible form, used whenever a circle is centred at the origin.

Reading off the centre and radius. Conversely, given an equation already in the form $(x-h)^2+(y-k)^2=r^2$, the centre and radius can be read off directly: the centre is $(h,k)$, and the radius is $r$ (never $r^2$ -- a common point of confusion, since the right-hand side is the radius squared).

Worked check. For a circle with centre $(2,-3)$ and radius $5$, the standard equation is $(x-2)^2+(y+3)^2=25$ -- note the sign flip: since the centre's $y$-coordinate is $-3$, the equation reads $(y-(-3))^2=(y+3)^2$. …